Then the angle required lies in the first quadrant and so= 68 Æ
to the nearest degree. 5 sin x + 2 cos x =
29 cos (x - 68)Æ
The same technique works with negative values of a and b. The difference lies only in determining the quadrant in which the answer lies.
Wave function exercise
30 min
There is a web exercise if you prefer it.
Q12: Express 4 sin x + cos x in the form k cos (x -) where k0 and 0360 Æ
Q13: Express 6 sin x + 3 cos x in the form k cos (x +) where
k0 and 0360 Æ
Q14: Express 3 sin x - 2 cos x in the form k sin (x -) where k0 and 0360 Æ
Q15: Express -sin x - 6 cos x in the form k sin (x +) where k0 and 0360 Æ
Q16: Express 2 cos x - 4 sin x in the form k cos (x -) where k0 and 0360 Æ
Q17: Express -
3 sin x + 2 cos x in the form k sin (x -)
where k0 and 0360 Æ
4.4
Maxima/minima values and solving equations
4.4.1 Maxima/minima values of the expression a sin x + b cos x
Æ
Learning Objective
Find maximum and minimum values of expressions of the form asin x + bcos x = c From the work covered in this topic, it is now straightforward to express
a sin x + b cos x in terms of a single cosine or sine function. The form of this new function has two variables say k and, although it is important to realise that other symbols can
be used. (r instead of k andinstead ofare very common.)
The amplitude of this single combined function is the value k
By definition, the amplitude of a wave function is half of the distance between the maximum and the minimum of the wave.
This is in fact how k is determined but the graphs of the functions in this topic are all symmetrical about the x-axis making this calculation unnecessary.
It follows that for a straightforward combined wave function, the maximum value is k and the minimum value is - k
100 TOPIC 4. FURTHER TRIGONOMETRIC RELATIONSHIPS
Example If 3 sin x + cos x can be expressed as
10 sin (x + 18)Æ
, find the maximum and minimum values of the function.
Answer:
It has been stated that the maximum is k; that is the maximum is
10 Similarly the minimum is -k; that is the minimum is -
10 Examine the basic graph:
The maximum and minimum values are clearly the values k and the amplitude is
confirmed as half the distance between the maximum and the minimum. The maximum value occurs when sin (x + 18)Æ
is at its maximum. That is at the value 1: then the maximum of
10 sin (x + 18)Æ
is
10 The minimum occurs when sin (x + 18)Æ
has the value -1: that is the minimum of
10 sin (x + 18)Æ
is -
10
From this it is possible to determine the maximum and minimum values for a variety of expressions.
Example What is the maximum and minimum values of the expression 4 + 5 cos (x -
37)Æ
? Answer:
The maximum is 4 + 5 = 9 and occurs when cos (x - 37)Æ
= 1 The minimum is 4 - 5 = -1 and occurs when cos (x - 37)Æ
= -1
Some care is needed though when the expression is slightly different.
Example What is the maximum and minimum values of the expression 3 - 7 sin (x -
20)Æ
? Answer:
In this case the maximum occurs when sin (x - 20)Æ
= -1as this gives a maximum of 3 - (-7) = 10
4.4. MAXIMA/MINIMA VALUES AND SOLVING EQUATIONS 101
It follows that the minimum occurs when sin (x - 20)Æ
= 1 as this gives 3 - 7 = -4
Using the techniques of this topic it is now possible to find the maximum and minimum of expressions such as -2 + 3 sin x - 5 cos x and the value of x at which these maxima and minima occur.
The general technique is as follows.
First of all express 3 sin x - 5 cos x as a combined trig function.
Secondly use this new function to determine the maximum and minimum values using the knowledge of max/min values of sin and cos functions.
Solve the single expression in sin or cos =1 to find the appropriate values of x
The first exercise will give practice in finding maximum and minimum values.
If only the maximum or minimum values are required, there is a shortcut. Since any of the four combined trig functions has the same maximum of 1 and minimum of -1 it follows that it is only the amplitude ( the value of k) which needs to be calculated.
Example What is the maximum and minimum values of the expression - 3 sin x + 4 + 5
cos x? Answer:
Isolate the combination of trig functions: -3 sin x + 5 cos x Here in general terms a = -3 and b = 5
But previously it was shown that k =
(a2+ b2)
In this case therefore, k =
(9 + 25) =
34
Returning to the original expression, - 3 sin x + 4 + 5 cos x = 4 + (-3 sin x + 5 cos x) If the combined expression for -3 sin x + 5 cos x is X then 4 +
34X will give a maximum at 4 +
34 when X = 1 a minimum at 4 -
34 when X = -1 Maximum and minimum exercise
30 min
There is another exercise on the web if you prefer it.
Q18: Find the maximum and minimum values of the following expressions:
a) 2 - 4 sin (x - 300)Æ b) -5 - cos (x + 34)Æ c) 5 - 5 sin (x - 30)Æ d) 3 + 2 sin x - 4 cos x e) -1 - cos x + sin x f) 2 sin x - 2 + cos x
102 TOPIC 4. FURTHER TRIGONOMETRIC RELATIONSHIPS 4.4.2 Solving equations Æ Learning Objective
Find the values of x for which maximum and minimum values of expressions of the form asin x + bcos x = c occur
The technique is best shown by example.
Example Solve the equation 2 sin x - 5 cos x = 4, 0x360 Æ
Answer:
2 sin x - 5 cos x can be expressed as say, k sin (x +)
2 sin x - 5 cos x = k sin x cos + k cos x sin
Equate the coefficients of sin x: 2 = k cos:- equation 1
Equate the coefficients of cos x: -5 = k sin:- equation 2
Squaring and adding gives 4 + 25 = k2(cos2+ sin
2 ) = k 2 k = 29
Dividing gives tan=
-5/ 2
In the first quadrant this solves to give= 68 Æ
to the nearest degree.
from equation 1: cos is positive from equation 2: sin is negative
The angle lies in quadrant four.
= 360 - 68 Æ
= 292Æ
2 sin x - 5 cos x can be expressed as
29 sin (x + 292)Æ So 2 sin x - 5 cos x = 29 sin (x + 292)Æ = 4 sin (x + 292)Æ = 4 29
Thus the first quadrant angle= 48 Æ
to the nearest degree. Since sin is positive the solutions lie in quadrants one and two.
4.5. SOLVING PROBLEMS WITH TRIG. FORMULAE 103